LMFDB - Embedded newform 4114.2.a.bg.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4114,2,Mod(1,4114)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4114, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4114.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4114 = 2 \cdot 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4114.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.8504553916$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 12x^{6} + 28x^{5} + 51x^{4} - 80x^{3} - 92x^{2} + 67x + 59 $ x^8 - 3x^7 - 12x^6 + 28x^5 + 51x^4 - 80x^3 - 92x^2 + 67*x + 59
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 374)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.87555$ of defining polynomial
Character	$\chi$	$=$	4114.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.875550 q^{3} +1.00000 q^{4} -0.120305 q^{5} -0.875550 q^{6} -1.45888 q^{7} -1.00000 q^{8} -2.23341 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.875550 q^{3} +1.00000 q^{4} -0.120305 q^{5} -0.875550 q^{6} -1.45888 q^{7} -1.00000 q^{8} -2.23341 q^{9} +0.120305 q^{10} +0.875550 q^{12} -0.132640 q^{13} +1.45888 q^{14} -0.105333 q^{15} +1.00000 q^{16} +1.00000 q^{17} +2.23341 q^{18} -2.69547 q^{19} -0.120305 q^{20} -1.27732 q^{21} +7.64311 q^{23} -0.875550 q^{24} -4.98553 q^{25} +0.132640 q^{26} -4.58211 q^{27} -1.45888 q^{28} +2.62915 q^{29} +0.105333 q^{30} +9.49616 q^{31} -1.00000 q^{32} -1.00000 q^{34} +0.175510 q^{35} -2.23341 q^{36} +9.39539 q^{37} +2.69547 q^{38} -0.116133 q^{39} +0.120305 q^{40} -9.27509 q^{41} +1.27732 q^{42} +1.52139 q^{43} +0.268690 q^{45} -7.64311 q^{46} +7.60585 q^{47} +0.875550 q^{48} -4.87167 q^{49} +4.98553 q^{50} +0.875550 q^{51} -0.132640 q^{52} -10.4897 q^{53} +4.58211 q^{54} +1.45888 q^{56} -2.36002 q^{57} -2.62915 q^{58} +0.648751 q^{59} -0.105333 q^{60} -13.8480 q^{61} -9.49616 q^{62} +3.25828 q^{63} +1.00000 q^{64} +0.0159572 q^{65} -5.15478 q^{67} +1.00000 q^{68} +6.69193 q^{69} -0.175510 q^{70} +11.0472 q^{71} +2.23341 q^{72} -1.39682 q^{73} -9.39539 q^{74} -4.36508 q^{75} -2.69547 q^{76} +0.116133 q^{78} -4.75388 q^{79} -0.120305 q^{80} +2.68836 q^{81} +9.27509 q^{82} -12.7972 q^{83} -1.27732 q^{84} -0.120305 q^{85} -1.52139 q^{86} +2.30196 q^{87} -3.91389 q^{89} -0.268690 q^{90} +0.193506 q^{91} +7.64311 q^{92} +8.31437 q^{93} -7.60585 q^{94} +0.324278 q^{95} -0.875550 q^{96} -7.50460 q^{97} +4.87167 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9	$ 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9} - 2 q^{10} - 5 q^{12} + 5 q^{13} + 11 q^{14} + 2 q^{15} + 8 q^{16} + 8 q^{17} - 11 q^{18} - 15 q^{19} + 2 q^{20} - 12 q^{23} + 5 q^{24} + 24 q^{25} - 5 q^{26} - 17 q^{27} - 11 q^{28} - 22 q^{29} - 2 q^{30} - 7 q^{31} - 8 q^{32} - 8 q^{34} - 10 q^{35} + 11 q^{36} + 15 q^{37} + 15 q^{38} - 35 q^{39} - 2 q^{40} - 17 q^{41} - 8 q^{43} + 3 q^{45} + 12 q^{46} + 18 q^{47} - 5 q^{48} + q^{49} - 24 q^{50} - 5 q^{51} + 5 q^{52} + 20 q^{53} + 17 q^{54} + 11 q^{56} - 9 q^{57} + 22 q^{58} + 6 q^{59} + 2 q^{60} + 5 q^{61} + 7 q^{62} + 14 q^{63} + 8 q^{64} - 3 q^{65} + 13 q^{67} + 8 q^{68} + 46 q^{69} + 10 q^{70} - 18 q^{71} - 11 q^{72} + 4 q^{73} - 15 q^{74} - 49 q^{75} - 15 q^{76} + 35 q^{78} - 21 q^{79} + 2 q^{80} + 8 q^{81} + 17 q^{82} - 38 q^{83} + 2 q^{85} + 8 q^{86} + 46 q^{87} + 12 q^{89} - 3 q^{90} - 5 q^{91} - 12 q^{92} + 3 q^{93} - 18 q^{94} - 63 q^{95} + 5 q^{96} - 66 q^{97} - q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9 - 2 * q^10 - 5 * q^12 + 5 * q^13 + 11 * q^14 + 2 * q^15 + 8 * q^16 + 8 * q^17 - 11 * q^18 - 15 * q^19 + 2 * q^20 - 12 * q^23 + 5 * q^24 + 24 * q^25 - 5 * q^26 - 17 * q^27 - 11 * q^28 - 22 * q^29 - 2 * q^30 - 7 * q^31 - 8 * q^32 - 8 * q^34 - 10 * q^35 + 11 * q^36 + 15 * q^37 + 15 * q^38 - 35 * q^39 - 2 * q^40 - 17 * q^41 - 8 * q^43 + 3 * q^45 + 12 * q^46 + 18 * q^47 - 5 * q^48 + q^49 - 24 * q^50 - 5 * q^51 + 5 * q^52 + 20 * q^53 + 17 * q^54 + 11 * q^56 - 9 * q^57 + 22 * q^58 + 6 * q^59 + 2 * q^60 + 5 * q^61 + 7 * q^62 + 14 * q^63 + 8 * q^64 - 3 * q^65 + 13 * q^67 + 8 * q^68 + 46 * q^69 + 10 * q^70 - 18 * q^71 - 11 * q^72 + 4 * q^73 - 15 * q^74 - 49 * q^75 - 15 * q^76 + 35 * q^78 - 21 * q^79 + 2 * q^80 + 8 * q^81 + 17 * q^82 - 38 * q^83 + 2 * q^85 + 8 * q^86 + 46 * q^87 + 12 * q^89 - 3 * q^90 - 5 * q^91 - 12 * q^92 + 3 * q^93 - 18 * q^94 - 63 * q^95 + 5 * q^96 - 66 * q^97 - q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.875550	0.505499	0.252750	−	0.967532i	$-0.418665\pi$
$3$	0.875550	0.505499	0.252750	+	0.967532i	$0.418665\pi$
$4$	1.00000	0.500000
$5$	−0.120305	−0.0538019	−0.0269009	−	0.999638i	$-0.508564\pi$
$5$	−0.120305	−0.0538019	−0.0269009	+	0.999638i	$0.508564\pi$
$6$	−0.875550	−0.357442
$7$	−1.45888	−0.551405	−0.275702	−	0.961243i	$-0.588910\pi$
$7$	−1.45888	−0.551405	−0.275702	+	0.961243i	$0.588910\pi$
$8$	−1.00000	−0.353553
$9$	−2.23341	−0.744471
$10$	0.120305	0.0380437
$11$	0	0
$11$	0	0
$12$	0.875550	0.252750
$13$	−0.132640	−0.0367878	−0.0183939	−	0.999831i	$-0.505855\pi$
$13$	−0.132640	−0.0367878	−0.0183939	+	0.999831i	$0.505855\pi$
$14$	1.45888	0.389902
$15$	−0.105333	−0.0271968
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	2.23341	0.526420
$19$	−2.69547	−0.618383	−0.309191	−	0.951000i	$-0.600058\pi$
$19$	−2.69547	−0.618383	−0.309191	+	0.951000i	$0.600058\pi$
$20$	−0.120305	−0.0269009
$21$	−1.27732	−0.278735
$22$	0	0
$23$	7.64311	1.59370	0.796850	−	0.604177i	$-0.206498\pi$
$23$	7.64311	1.59370	0.796850	+	0.604177i	$0.206498\pi$
$24$	−0.875550	−0.178721
$25$	−4.98553	−0.997105
$26$	0.132640	0.0260129
$27$	−4.58211	−0.881828
$28$	−1.45888	−0.275702
$29$	2.62915	0.488222	0.244111	−	0.969747i	$-0.421504\pi$
$29$	2.62915	0.488222	0.244111	+	0.969747i	$0.421504\pi$
$30$	0.105333	0.0192311
$31$	9.49616	1.70556	0.852781	−	0.522269i	$-0.174914\pi$
$31$	9.49616	1.70556	0.852781	+	0.522269i	$0.174914\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0.175510	0.0296666
$36$	−2.23341	−0.372235
$37$	9.39539	1.54459	0.772296	−	0.635262i	$-0.219108\pi$
$37$	9.39539	1.54459	0.772296	+	0.635262i	$0.219108\pi$
$38$	2.69547	0.437263
$39$	−0.116133	−0.0185962
$40$	0.120305	0.0190218
$41$	−9.27509	−1.44853	−0.724263	−	0.689524i	$-0.757820\pi$
$41$	−9.27509	−1.44853	−0.724263	+	0.689524i	$0.757820\pi$
$42$	1.27732	0.197095
$43$	1.52139	0.232010	0.116005	−	0.993249i	$-0.462991\pi$
$43$	1.52139	0.232010	0.116005	+	0.993249i	$0.462991\pi$
$44$	0	0
$45$	0.268690	0.0400539
$46$	−7.64311	−1.12692
$47$	7.60585	1.10943	0.554714	−	0.832041i	$-0.312828\pi$
$47$	7.60585	1.10943	0.554714	+	0.832041i	$0.312828\pi$
$48$	0.875550	0.126375
$49$	−4.87167	−0.695953
$50$	4.98553	0.705060
$51$	0.875550	0.122602
$52$	−0.132640	−0.0183939
$53$	−10.4897	−1.44087	−0.720437	−	0.693521i	$-0.756059\pi$
$53$	−10.4897	−1.44087	−0.720437	+	0.693521i	$0.756059\pi$
$54$	4.58211	0.623547
$55$	0	0
$56$	1.45888	0.194951
$57$	−2.36002	−0.312592
$58$	−2.62915	−0.345225
$59$	0.648751	0.0844602	0.0422301	−	0.999108i	$-0.486554\pi$
$59$	0.648751	0.0844602	0.0422301	+	0.999108i	$0.486554\pi$
$60$	−0.105333	−0.0135984
$61$	−13.8480	−1.77305	−0.886527	−	0.462677i	$-0.846889\pi$
$61$	−13.8480	−1.77305	−0.886527	+	0.462677i	$0.846889\pi$
$62$	−9.49616	−1.20601
$63$	3.25828	0.410505
$64$	1.00000	0.125000
$65$	0.0159572	0.00197925
$66$	0	0
$67$	−5.15478	−0.629757	−0.314878	−	0.949132i	$-0.601964\pi$
$67$	−5.15478	−0.629757	−0.314878	+	0.949132i	$0.601964\pi$
$68$	1.00000	0.121268
$69$	6.69193	0.805614
$70$	−0.175510	−0.0209775
$71$	11.0472	1.31106	0.655530	−	0.755169i	$-0.272445\pi$
$71$	11.0472	1.31106	0.655530	+	0.755169i	$0.272445\pi$
$72$	2.23341	0.263210
$73$	−1.39682	−0.163485	−0.0817424	−	0.996653i	$-0.526048\pi$
$73$	−1.39682	−0.163485	−0.0817424	+	0.996653i	$0.526048\pi$
$74$	−9.39539	−1.09219
$75$	−4.36508	−0.504036
$76$	−2.69547	−0.309191
$77$	0	0
$78$	0.116133	0.0131495
$79$	−4.75388	−0.534854	−0.267427	−	0.963578i	$-0.586173\pi$
$79$	−4.75388	−0.534854	−0.267427	+	0.963578i	$0.586173\pi$
$80$	−0.120305	−0.0134505
$81$	2.68836	0.298707
$82$	9.27509	1.02426
$83$	−12.7972	−1.40468	−0.702338	−	0.711844i	$-0.747860\pi$
$83$	−12.7972	−1.40468	−0.702338	+	0.711844i	$0.747860\pi$
$84$	−1.27732	−0.139367
$85$	−0.120305	−0.0130489
$86$	−1.52139	−0.164056
$87$	2.30196	0.246796
$88$	0	0
$89$	−3.91389	−0.414871	−0.207436	−	0.978249i	$-0.566512\pi$
$89$	−3.91389	−0.414871	−0.207436	+	0.978249i	$0.566512\pi$
$90$	−0.268690	−0.0283224
$91$	0.193506	0.0202850
$92$	7.64311	0.796850
$93$	8.31437	0.862160
$94$	−7.60585	−0.784483
$95$	0.324278	0.0332702
$96$	−0.875550	−0.0893605
$97$	−7.50460	−0.761977	−0.380988	−	0.924580i	$-0.624416\pi$
$97$	−7.50460	−0.761977	−0.380988	+	0.924580i	$0.624416\pi$
$98$	4.87167	0.492113
$99$	0	0
$100$	−4.98553	−0.498553
$101$	0.602898	0.0599905	0.0299953	−	0.999550i	$-0.490451\pi$
$101$	0.602898	0.0599905	0.0299953	+	0.999550i	$0.490451\pi$
$102$	−0.875550	−0.0866924
$103$	−16.2522	−1.60137	−0.800687	−	0.599083i	$-0.795532\pi$
$103$	−16.2522	−1.60137	−0.800687	+	0.599083i	$0.795532\pi$
$104$	0.132640	0.0130064
$105$	0.153668	0.0149965
$106$	10.4897	1.01885
$107$	−19.8815	−1.92202	−0.961010	−	0.276514i	$-0.910821\pi$
$107$	−19.8815	−1.92202	−0.961010	+	0.276514i	$0.910821\pi$
$108$	−4.58211	−0.440914
$109$	18.5853	1.78014	0.890072	−	0.455819i	$-0.150654\pi$
$109$	18.5853	1.78014	0.890072	+	0.455819i	$0.150654\pi$
$110$	0	0
$111$	8.22614	0.780790
$112$	−1.45888	−0.137851
$113$	−18.3469	−1.72593	−0.862964	−	0.505265i	$-0.831395\pi$
$113$	−18.3469	−1.72593	−0.862964	+	0.505265i	$0.831395\pi$
$114$	2.36002	0.221036
$115$	−0.919503	−0.0857441
$116$	2.62915	0.244111
$117$	0.296240	0.0273874
$118$	−0.648751	−0.0597224
$119$	−1.45888	−0.133735
$120$	0.105333	0.00961553
$121$	0	0
$122$	13.8480	1.25374
$123$	−8.12080	−0.732228
$124$	9.49616	0.852781
$125$	1.20131	0.107448
$126$	−3.25828	−0.290271
$127$	−8.74115	−0.775651	−0.387826	−	0.921733i	$-0.626774\pi$
$127$	−8.74115	−0.775651	−0.387826	+	0.921733i	$0.626774\pi$
$128$	−1.00000	−0.0883883
$129$	1.33205	0.117281
$130$	−0.0159572	−0.00139954
$131$	−2.90096	−0.253458	−0.126729	−	0.991937i	$-0.540448\pi$
$131$	−2.90096	−0.253458	−0.126729	+	0.991937i	$0.540448\pi$
$132$	0	0
$133$	3.93237	0.340979
$134$	5.15478	0.445305
$135$	0.551250	0.0474440
$136$	−1.00000	−0.0857493
$137$	−19.6664	−1.68021	−0.840106	−	0.542423i	$-0.817507\pi$
$137$	−19.6664	−1.68021	−0.840106	+	0.542423i	$0.817507\pi$
$138$	−6.69193	−0.569655
$139$	−1.84809	−0.156753	−0.0783765	−	0.996924i	$-0.524974\pi$
$139$	−1.84809	−0.156753	−0.0783765	+	0.996924i	$0.524974\pi$
$140$	0.175510	0.0148333
$141$	6.65930	0.560814
$142$	−11.0472	−0.927060
$143$	0	0
$144$	−2.23341	−0.186118
$145$	−0.316300	−0.0262673
$146$	1.39682	0.115601
$147$	−4.26539	−0.351803
$148$	9.39539	0.772296
$149$	−2.85559	−0.233939	−0.116970	−	0.993135i	$-0.537318\pi$
$149$	−2.85559	−0.233939	−0.116970	+	0.993135i	$0.537318\pi$
$150$	4.36508	0.356407
$151$	−19.2777	−1.56880	−0.784400	−	0.620255i	$-0.787029\pi$
$151$	−19.2777	−1.56880	−0.784400	+	0.620255i	$0.787029\pi$
$152$	2.69547	0.218631
$153$	−2.23341	−0.180561
$154$	0	0
$155$	−1.14243	−0.0917624
$156$	−0.116133	−0.00929809
$157$	5.36233	0.427961	0.213980	−	0.976838i	$-0.431357\pi$
$157$	5.36233	0.427961	0.213980	+	0.976838i	$0.431357\pi$
$158$	4.75388	0.378199
$159$	−9.18427	−0.728360
$160$	0.120305	0.00951092
$161$	−11.1504	−0.878774
$162$	−2.68836	−0.211218
$163$	−8.15707	−0.638911	−0.319456	−	0.947601i	$-0.603500\pi$
$163$	−8.15707	−0.638911	−0.319456	+	0.947601i	$0.603500\pi$
$164$	−9.27509	−0.724263
$165$	0	0
$166$	12.7972	0.993256
$167$	−8.23508	−0.637250	−0.318625	−	0.947881i	$-0.603221\pi$
$167$	−8.23508	−0.637250	−0.318625	+	0.947881i	$0.603221\pi$
$168$	1.27732	0.0985476
$169$	−12.9824	−0.998647
$170$	0.120305	0.00922695
$171$	6.02009	0.460368
$172$	1.52139	0.116005
$173$	−4.06308	−0.308911	−0.154455	−	0.988000i	$-0.549362\pi$
$173$	−4.06308	−0.308911	−0.154455	+	0.988000i	$0.549362\pi$
$174$	−2.30196	−0.174511
$175$	7.27329	0.549809
$176$	0	0
$177$	0.568014	0.0426945
$178$	3.91389	0.293358
$179$	7.12558	0.532591	0.266295	−	0.963891i	$-0.414200\pi$
$179$	7.12558	0.532591	0.266295	+	0.963891i	$0.414200\pi$
$180$	0.268690	0.0200270
$181$	7.33962	0.545550	0.272775	−	0.962078i	$-0.412059\pi$
$181$	7.33962	0.545550	0.272775	+	0.962078i	$0.412059\pi$
$182$	−0.193506	−0.0143436
$183$	−12.1246	−0.896277
$184$	−7.64311	−0.563458
$185$	−1.13031	−0.0831020
$186$	−8.31437	−0.609639
$187$	0	0
$188$	7.60585	0.554714
$189$	6.68476	0.486244
$190$	−0.324278	−0.0235256
$191$	−8.04902	−0.582407	−0.291203	−	0.956661i	$-0.594056\pi$
$191$	−8.04902	−0.582407	−0.291203	+	0.956661i	$0.594056\pi$
$192$	0.875550	0.0631874
$193$	11.8523	0.853147	0.426574	−	0.904453i	$-0.359721\pi$
$193$	11.8523	0.853147	0.426574	+	0.904453i	$0.359721\pi$
$194$	7.50460	0.538799
$195$	0.0139714	0.00100051
$196$	−4.87167	−0.347976
$197$	8.46442	0.603065	0.301533	−	0.953456i	$-0.402502\pi$
$197$	8.46442	0.603065	0.301533	+	0.953456i	$0.402502\pi$
$198$	0	0
$199$	−3.17230	−0.224879	−0.112439	−	0.993659i	$-0.535866\pi$
$199$	−3.17230	−0.224879	−0.112439	+	0.993659i	$0.535866\pi$
$200$	4.98553	0.352530
$201$	−4.51327	−0.318342
$202$	−0.602898	−0.0424197
$203$	−3.83562	−0.269208
$204$	0.875550	0.0613008
$205$	1.11584	0.0779334
$206$	16.2522	1.13234
$207$	−17.0702	−1.18646
$208$	−0.132640	−0.00919694
$209$	0	0
$210$	−0.153668	−0.0106041
$211$	11.6659	0.803111	0.401555	−	0.915835i	$-0.368470\pi$
$211$	11.6659	0.803111	0.401555	+	0.915835i	$0.368470\pi$
$212$	−10.4897	−0.720437
$213$	9.67237	0.662740
$214$	19.8815	1.35907
$215$	−0.183031	−0.0124826
$216$	4.58211	0.311773
$217$	−13.8538	−0.940455
$218$	−18.5853	−1.25875
$219$	−1.22298	−0.0826415
$220$	0	0
$221$	−0.132640	−0.00892234
$222$	−8.22614	−0.552102
$223$	4.25903	0.285206	0.142603	−	0.989780i	$-0.454453\pi$
$223$	4.25903	0.285206	0.142603	+	0.989780i	$0.454453\pi$
$224$	1.45888	0.0974755
$225$	11.1347	0.742316
$226$	18.3469	1.22042
$227$	19.7379	1.31005	0.655026	−	0.755606i	$-0.272658\pi$
$227$	19.7379	1.31005	0.655026	+	0.755606i	$0.272658\pi$
$228$	−2.36002	−0.156296
$229$	0.573514	0.0378989	0.0189494	−	0.999820i	$-0.493968\pi$
$229$	0.573514	0.0378989	0.0189494	+	0.999820i	$0.493968\pi$
$230$	0.919503	0.0606302
$231$	0	0
$232$	−2.62915	−0.172612
$233$	10.5622	0.691950	0.345975	−	0.938244i	$-0.387548\pi$
$233$	10.5622	0.691950	0.345975	+	0.938244i	$0.387548\pi$
$234$	−0.296240	−0.0193658
$235$	−0.915019	−0.0596893
$236$	0.648751	0.0422301
$237$	−4.16226	−0.270368
$238$	1.45888	0.0945652
$239$	−19.5572	−1.26505	−0.632524	−	0.774541i	$-0.717981\pi$
$239$	−19.5572	−1.26505	−0.632524	+	0.774541i	$0.717981\pi$
$240$	−0.105333	−0.00679920
$241$	20.0960	1.29450	0.647248	−	0.762280i	$-0.275920\pi$
$241$	20.0960	1.29450	0.647248	+	0.762280i	$0.275920\pi$
$242$	0	0
$243$	16.1001	1.03282
$244$	−13.8480	−0.886527
$245$	0.586085	0.0374436
$246$	8.12080	0.517764
$247$	0.357527	0.0227489
$248$	−9.49616	−0.603007
$249$	−11.2046	−0.710062
$250$	−1.20131	−0.0759772
$251$	−2.85046	−0.179920	−0.0899598	−	0.995945i	$-0.528674\pi$
$251$	−2.85046	−0.179920	−0.0899598	+	0.995945i	$0.528674\pi$
$252$	3.25828	0.205252
$253$	0	0
$254$	8.74115	0.548468
$255$	−0.105333	−0.00659620
$256$	1.00000	0.0625000
$257$	−20.2888	−1.26558	−0.632790	−	0.774324i	$-0.718090\pi$
$257$	−20.2888	−1.26558	−0.632790	+	0.774324i	$0.718090\pi$
$258$	−1.33205	−0.0829301
$259$	−13.7068	−0.851696
$260$	0.0159572	0.000989626 0
$261$	−5.87198	−0.363467
$262$	2.90096	0.179222
$263$	20.8656	1.28663	0.643315	−	0.765602i	$-0.277559\pi$
$263$	20.8656	1.28663	0.643315	+	0.765602i	$0.277559\pi$
$264$	0	0
$265$	1.26196	0.0775217
$266$	−3.93237	−0.241109
$267$	−3.42681	−0.209717
$268$	−5.15478	−0.314878
$269$	16.0030	0.975722	0.487861	−	0.872921i	$-0.337777\pi$
$269$	16.0030	0.975722	0.487861	+	0.872921i	$0.337777\pi$
$270$	−0.551250	−0.0335480
$271$	22.7203	1.38016	0.690079	−	0.723734i	$-0.257576\pi$
$271$	22.7203	1.38016	0.690079	+	0.723734i	$0.257576\pi$
$272$	1.00000	0.0606339
$273$	0.169424	0.0102540
$274$	19.6664	1.18809
$275$	0	0
$276$	6.69193	0.402807
$277$	−12.5533	−0.754256	−0.377128	−	0.926161i	$-0.623088\pi$
$277$	−12.5533	−0.754256	−0.377128	+	0.926161i	$0.623088\pi$
$278$	1.84809	0.110841
$279$	−21.2088	−1.26974
$280$	−0.175510	−0.0104887
$281$	7.99925	0.477195	0.238597	−	0.971119i	$-0.423312\pi$
$281$	7.99925	0.477195	0.238597	+	0.971119i	$0.423312\pi$
$282$	−6.65930	−0.396556
$283$	26.5919	1.58073	0.790363	−	0.612639i	$-0.209892\pi$
$283$	26.5919	1.58073	0.790363	+	0.612639i	$0.209892\pi$
$284$	11.0472	0.655530
$285$	0.283921	0.0168180
$286$	0	0
$287$	13.5312	0.798724
$288$	2.23341	0.131605
$289$	1.00000	0.0588235
$290$	0.316300	0.0185738
$291$	−6.57066	−0.385179
$292$	−1.39682	−0.0817424
$293$	−30.1438	−1.76102	−0.880509	−	0.474029i	$-0.842799\pi$
$293$	−30.1438	−1.76102	−0.880509	+	0.474029i	$0.842799\pi$
$294$	4.26539	0.248763
$295$	−0.0780478	−0.00454412
$296$	−9.39539	−0.546096
$297$	0	0
$298$	2.85559	0.165420
$299$	−1.01378	−0.0586286
$300$	−4.36508	−0.252018
$301$	−2.21953	−0.127931
$302$	19.2777	1.10931
$303$	0.527867	0.0303252
$304$	−2.69547	−0.154596
$305$	1.66598	0.0953937
$306$	2.23341	0.127676
$307$	−15.1612	−0.865297	−0.432649	−	0.901563i	$-0.642421\pi$
$307$	−15.1612	−0.865297	−0.432649	+	0.901563i	$0.642421\pi$
$308$	0	0
$309$	−14.2296	−0.809493
$310$	1.14243	0.0648858
$311$	−1.98848	−0.112756	−0.0563781	−	0.998409i	$-0.517955\pi$
$311$	−1.98848	−0.112756	−0.0563781	+	0.998409i	$0.517955\pi$
$312$	0.116133	0.00657474
$313$	20.1505	1.13897	0.569486	−	0.822001i	$-0.307142\pi$
$313$	20.1505	1.13897	0.569486	+	0.822001i	$0.307142\pi$
$314$	−5.36233	−0.302614
$315$	−0.391986	−0.0220859
$316$	−4.75388	−0.267427
$317$	16.2445	0.912380	0.456190	−	0.889882i	$-0.349214\pi$
$317$	16.2445	0.912380	0.456190	+	0.889882i	$0.349214\pi$
$318$	9.18427	0.515028
$319$	0	0
$320$	−0.120305	−0.00672524
$321$	−17.4073	−0.971579
$322$	11.1504	0.621387
$323$	−2.69547	−0.149980
$324$	2.68836	0.149354
$325$	0.661281	0.0366813
$326$	8.15707	0.451779
$327$	16.2723	0.899861
$328$	9.27509	0.512131
$329$	−11.0960	−0.611744
$330$	0	0
$331$	−16.0144	−0.880232	−0.440116	−	0.897941i	$-0.645063\pi$
$331$	−16.0144	−0.880232	−0.440116	+	0.897941i	$0.645063\pi$
$332$	−12.7972	−0.702338
$333$	−20.9838	−1.14990
$334$	8.23508	0.450604
$335$	0.620144	0.0338821
$336$	−1.27732	−0.0696837
$337$	−1.23076	−0.0670437	−0.0335218	−	0.999438i	$-0.510672\pi$
$337$	−1.23076	−0.0670437	−0.0335218	+	0.999438i	$0.510672\pi$
$338$	12.9824	0.706150
$339$	−16.0636	−0.872455
$340$	−0.120305	−0.00652444
$341$	0	0
$342$	−6.02009	−0.325529
$343$	17.3193	0.935157
$344$	−1.52139	−0.0820279
$345$	−0.805071	−0.0433435
$346$	4.06308	0.218433
$347$	−21.1460	−1.13518	−0.567588	−	0.823313i	$-0.692123\pi$
$347$	−21.1460	−1.13518	−0.567588	+	0.823313i	$0.692123\pi$
$348$	2.30196	0.123398
$349$	−9.48933	−0.507952	−0.253976	−	0.967210i	$-0.581738\pi$
$349$	−9.48933	−0.507952	−0.253976	+	0.967210i	$0.581738\pi$
$350$	−7.27329	−0.388774
$351$	0.607772	0.0324405
$352$	0	0
$353$	1.34305	0.0714833	0.0357417	−	0.999361i	$-0.488621\pi$
$353$	1.34305	0.0714833	0.0357417	+	0.999361i	$0.488621\pi$
$354$	−0.568014	−0.0301896
$355$	−1.32903	−0.0705375
$356$	−3.91389	−0.207436
$357$	−1.27732	−0.0676031
$358$	−7.12558	−0.376599
$359$	−25.7041	−1.35661	−0.678306	−	0.734779i	$-0.737286\pi$
$359$	−25.7041	−1.35661	−0.678306	+	0.734779i	$0.737286\pi$
$360$	−0.268690	−0.0141612
$361$	−11.7344	−0.617603
$362$	−7.33962	−0.385762
$363$	0	0
$364$	0.193506	0.0101425
$365$	0.168043	0.00879580
$366$	12.1246	0.633764
$367$	25.0115	1.30559	0.652796	−	0.757534i	$-0.273596\pi$
$367$	25.0115	1.30559	0.652796	+	0.757534i	$0.273596\pi$
$368$	7.64311	0.398425
$369$	20.7151	1.07838
$370$	1.13031	0.0587620
$371$	15.3032	0.794505
$372$	8.31437	0.431080
$373$	−5.43001	−0.281155	−0.140578	−	0.990070i	$-0.544896\pi$
$373$	−5.43001	−0.281155	−0.140578	+	0.990070i	$0.544896\pi$
$374$	0	0
$375$	1.05180	0.0543149
$376$	−7.60585	−0.392242
$377$	−0.348731	−0.0179606
$378$	−6.68476	−0.343827
$379$	−3.69485	−0.189792	−0.0948959	−	0.995487i	$-0.530252\pi$
$379$	−3.69485	−0.189792	−0.0948959	+	0.995487i	$0.530252\pi$
$380$	0.324278	0.0166351
$381$	−7.65331	−0.392091
$382$	8.04902	0.411824
$383$	−2.40475	−0.122877	−0.0614385	−	0.998111i	$-0.519569\pi$
$383$	−2.40475	−0.122877	−0.0614385	+	0.998111i	$0.519569\pi$
$384$	−0.875550	−0.0446802
$385$	0	0
$386$	−11.8523	−0.603266
$387$	−3.39789	−0.172725
$388$	−7.50460	−0.380988
$389$	−27.8761	−1.41337	−0.706687	−	0.707526i	$-0.749811\pi$
$389$	−27.8761	−1.41337	−0.706687	+	0.707526i	$0.749811\pi$
$390$	−0.0139714	−0.000707467 0
$391$	7.64311	0.386529
$392$	4.87167	0.246056
$393$	−2.53994	−0.128123
$394$	−8.46442	−0.426431
$395$	0.571914	0.0287761
$396$	0	0
$397$	−20.1130	−1.00944	−0.504722	−	0.863282i	$-0.668405\pi$
$397$	−20.1130	−1.00944	−0.504722	+	0.863282i	$0.668405\pi$
$398$	3.17230	0.159013
$399$	3.44298	0.172365
$400$	−4.98553	−0.249276
$401$	−6.79434	−0.339293	−0.169646	−	0.985505i	$-0.554263\pi$
$401$	−6.79434	−0.339293	−0.169646	+	0.985505i	$0.554263\pi$
$402$	4.51327	0.225101
$403$	−1.25957	−0.0627438
$404$	0.602898	0.0299953
$405$	−0.323423	−0.0160710
$406$	3.83562	0.190359
$407$	0	0
$408$	−0.875550	−0.0433462
$409$	16.8723	0.834283	0.417141	−	0.908842i	$-0.363032\pi$
$409$	16.8723	0.834283	0.417141	+	0.908842i	$0.363032\pi$
$410$	−1.11584	−0.0551072
$411$	−17.2189	−0.849345
$412$	−16.2522	−0.800687
$413$	−0.946450	−0.0465718
$414$	17.0702	0.838956
$415$	1.53956	0.0755742
$416$	0.132640	0.00650322
$417$	−1.61810	−0.0792385
$418$	0	0
$419$	−19.9568	−0.974953	−0.487476	−	0.873136i	$-0.662082\pi$
$419$	−19.9568	−0.974953	−0.487476	+	0.873136i	$0.662082\pi$
$420$	0.153668	0.00749823
$421$	−4.17192	−0.203327	−0.101663	−	0.994819i	$-0.532416\pi$
$421$	−4.17192	−0.203327	−0.101663	+	0.994819i	$0.532416\pi$
$422$	−11.6659	−0.567885
$423$	−16.9870	−0.825936
$424$	10.4897	0.509426
$425$	−4.98553	−0.241834
$426$	−9.67237	−0.468628
$427$	20.2026	0.977671
$428$	−19.8815	−0.961010
$429$	0	0
$430$	0.183031	0.00882652
$431$	−3.11158	−0.149879	−0.0749397	−	0.997188i	$-0.523876\pi$
$431$	−3.11158	−0.149879	−0.0749397	+	0.997188i	$0.523876\pi$
$432$	−4.58211	−0.220457
$433$	27.7967	1.33582	0.667912	−	0.744240i	$-0.267188\pi$
$433$	27.7967	1.33582	0.667912	+	0.744240i	$0.267188\pi$
$434$	13.8538	0.665002
$435$	−0.276936	−0.0132781
$436$	18.5853	0.890072
$437$	−20.6018	−0.985517
$438$	1.22298	0.0584363
$439$	−17.0281	−0.812705	−0.406353	−	0.913716i	$-0.633200\pi$
$439$	−17.0281	−0.812705	−0.406353	+	0.913716i	$0.633200\pi$
$440$	0	0
$441$	10.8804	0.518116
$442$	0.132640	0.00630905
$443$	22.8935	1.08770	0.543852	−	0.839181i	$-0.316965\pi$
$443$	22.8935	1.08770	0.543852	+	0.839181i	$0.316965\pi$
$444$	8.22614	0.390395
$445$	0.470859	0.0223209
$446$	−4.25903	−0.201671
$447$	−2.50021	−0.118256
$448$	−1.45888	−0.0689256
$449$	−10.1036	−0.476821	−0.238410	−	0.971165i	$-0.576626\pi$
$449$	−10.1036	−0.476821	−0.238410	+	0.971165i	$0.576626\pi$
$450$	−11.1347	−0.524896
$451$	0	0
$452$	−18.3469	−0.862964
$453$	−16.8786	−0.793027
$454$	−19.7379	−0.926347
$455$	−0.0232797	−0.00109137
$456$	2.36002	0.110518
$457$	41.7070	1.95097	0.975485	−	0.220064i	$-0.0706267\pi$
$457$	41.7070	1.95097	0.975485	+	0.220064i	$0.0706267\pi$
$458$	−0.573514	−0.0267986
$459$	−4.58211	−0.213875
$460$	−0.919503	−0.0428720
$461$	12.0145	0.559569	0.279785	−	0.960063i	$-0.409737\pi$
$461$	12.0145	0.559569	0.279785	+	0.960063i	$0.409737\pi$
$462$	0	0
$463$	−19.0604	−0.885813	−0.442907	−	0.896568i	$-0.646053\pi$
$463$	−19.0604	−0.885813	−0.442907	+	0.896568i	$0.646053\pi$
$464$	2.62915	0.122055
$465$	−1.00026	−0.0463858
$466$	−10.5622	−0.489283
$467$	−4.92280	−0.227800	−0.113900	−	0.993492i	$-0.536334\pi$
$467$	−4.92280	−0.227800	−0.113900	+	0.993492i	$0.536334\pi$
$468$	0.296240	0.0136937
$469$	7.52021	0.347251
$470$	0.915019	0.0422067
$471$	4.69499	0.216334
$472$	−0.648751	−0.0298612
$473$	0	0
$474$	4.16226	0.191179
$475$	13.4383	0.616593
$476$	−1.45888	−0.0668677
$477$	23.4279	1.07269
$478$	19.5572	0.894523
$479$	−31.8201	−1.45390	−0.726949	−	0.686691i	$-0.759062\pi$
$479$	−31.8201	−1.45390	−0.726949	+	0.686691i	$0.759062\pi$
$480$	0.105333	0.00480776
$481$	−1.24621	−0.0568221
$482$	−20.0960	−0.915346
$483$	−9.76273	−0.444219
$484$	0	0
$485$	0.902839	0.0409958
$486$	−16.1001	−0.730317
$487$	16.4732	0.746470	0.373235	−	0.927737i	$-0.378248\pi$
$487$	16.4732	0.746470	0.373235	+	0.927737i	$0.378248\pi$
$488$	13.8480	0.626869
$489$	−7.14193	−0.322969
$490$	−0.586085	−0.0264766
$491$	−35.2629	−1.59139	−0.795696	−	0.605696i	$-0.792895\pi$
$491$	−35.2629	−1.59139	−0.795696	+	0.605696i	$0.792895\pi$
$492$	−8.12080	−0.366114
$493$	2.62915	0.118411
$494$	−0.357527	−0.0160859
$495$	0	0
$496$	9.49616	0.426390
$497$	−16.1165	−0.722925
$498$	11.2046	0.502090
$499$	−31.9481	−1.43019	−0.715096	−	0.699026i	$-0.753617\pi$
$499$	−31.9481	−1.43019	−0.715096	+	0.699026i	$0.753617\pi$
$500$	1.20131	0.0537240
$501$	−7.21023	−0.322129
$502$	2.85046	0.127222
$503$	16.2955	0.726582	0.363291	−	0.931676i	$-0.381653\pi$
$503$	16.2955	0.726582	0.363291	+	0.931676i	$0.381653\pi$
$504$	−3.25828	−0.145135
$505$	−0.0725314	−0.00322761
$506$	0	0
$507$	−11.3667	−0.504815
$508$	−8.74115	−0.387826
$509$	1.78612	0.0791685	0.0395843	−	0.999216i	$-0.487397\pi$
$509$	1.78612	0.0791685	0.0395843	+	0.999216i	$0.487397\pi$
$510$	0.105333	0.00466421
$511$	2.03779	0.0901464
$512$	−1.00000	−0.0441942
$513$	12.3509	0.545308
$514$	20.2888	0.894900
$515$	1.95521	0.0861569
$516$	1.33205	0.0586404
$517$	0	0
$518$	13.7068	0.602240
$519$	−3.55743	−0.156154
$520$	−0.0159572	−0.000699771 0
$521$	−11.3857	−0.498815	−0.249408	−	0.968399i	$-0.580236\pi$
$521$	−11.3857	−0.498815	−0.249408	+	0.968399i	$0.580236\pi$
$522$	5.87198	0.257010
$523$	18.4908	0.808548	0.404274	−	0.914638i	$-0.367524\pi$
$523$	18.4908	0.808548	0.404274	+	0.914638i	$0.367524\pi$
$524$	−2.90096	−0.126729
$525$	6.36813	0.277928
$526$	−20.8656	−0.909785
$527$	9.49616	0.413659
$528$	0	0
$529$	35.4172	1.53988
$530$	−1.26196	−0.0548161
$531$	−1.44893	−0.0628781
$532$	3.93237	0.170490
$533$	1.23025	0.0532880
$534$	3.42681	0.148292
$535$	2.39184	0.103408
$536$	5.15478	0.222653
$537$	6.23880	0.269224
$538$	−16.0030	−0.689940
$539$	0	0
$540$	0.551250	0.0237220
$541$	24.5955	1.05745	0.528723	−	0.848795i	$-0.322671\pi$
$541$	24.5955	1.05745	0.528723	+	0.848795i	$0.322671\pi$
$542$	−22.7203	−0.975919
$543$	6.42621	0.275775
$544$	−1.00000	−0.0428746
$545$	−2.23589	−0.0957751
$546$	−0.169424	−0.00725069
$547$	7.33926	0.313804	0.156902	−	0.987614i	$-0.449849\pi$
$547$	7.33926	0.313804	0.156902	+	0.987614i	$0.449849\pi$
$548$	−19.6664	−0.840106
$549$	30.9283	1.31999
$550$	0	0
$551$	−7.08680	−0.301908
$552$	−6.69193	−0.284827
$553$	6.93535	0.294921
$554$	12.5533	0.533340
$555$	−0.989643	−0.0420080
$556$	−1.84809	−0.0783765
$557$	12.7140	0.538709	0.269354	−	0.963041i	$-0.413190\pi$
$557$	12.7140	0.538709	0.269354	+	0.963041i	$0.413190\pi$
$558$	21.2088	0.897842
$559$	−0.201798	−0.00853513
$560$	0.175510	0.00741666
$561$	0	0
$562$	−7.99925	−0.337428
$563$	14.0353	0.591518	0.295759	−	0.955263i	$-0.404427\pi$
$563$	14.0353	0.591518	0.295759	+	0.955263i	$0.404427\pi$
$564$	6.65930	0.280407
$565$	2.20721	0.0928582
$566$	−26.5919	−1.11774
$567$	−3.92200	−0.164709
$568$	−11.0472	−0.463530
$569$	−30.3143	−1.27084	−0.635421	−	0.772166i	$-0.719174\pi$
$569$	−30.3143	−1.27084	−0.635421	+	0.772166i	$0.719174\pi$
$570$	−0.283921	−0.0118922
$571$	−16.7495	−0.700944	−0.350472	−	0.936573i	$-0.613979\pi$
$571$	−16.7495	−0.700944	−0.350472	+	0.936573i	$0.613979\pi$
$572$	0	0
$573$	−7.04732	−0.294406
$574$	−13.5312	−0.564783
$575$	−38.1050	−1.58909
$576$	−2.23341	−0.0930588
$577$	−36.3764	−1.51437	−0.757185	−	0.653200i	$-0.773426\pi$
$577$	−36.3764	−1.51437	−0.757185	+	0.653200i	$0.773426\pi$
$578$	−1.00000	−0.0415945
$579$	10.3773	0.431265
$580$	−0.316300	−0.0131336
$581$	18.6696	0.774545
$582$	6.57066	0.272362
$583$	0	0
$584$	1.39682	0.0578006
$585$	−0.0356391	−0.00147349
$586$	30.1438	1.24523
$587$	−30.6253	−1.26404	−0.632021	−	0.774951i	$-0.717774\pi$
$587$	−30.6253	−1.26404	−0.632021	+	0.774951i	$0.717774\pi$
$588$	−4.26539	−0.175902
$589$	−25.5966	−1.05469
$590$	0.0780478	0.00321318
$591$	7.41103	0.304849
$592$	9.39539	0.386148
$593$	−31.8601	−1.30834	−0.654168	−	0.756349i	$-0.726981\pi$
$593$	−31.8601	−1.30834	−0.654168	+	0.756349i	$0.726981\pi$
$594$	0	0
$595$	0.175510	0.00719521
$596$	−2.85559	−0.116970
$597$	−2.77751	−0.113676
$598$	1.01378	0.0414567
$599$	−36.0861	−1.47444	−0.737219	−	0.675654i	$-0.763861\pi$
$599$	−36.0861	−1.47444	−0.737219	+	0.675654i	$0.763861\pi$
$600$	4.36508	0.178204
$601$	−17.0578	−0.695803	−0.347901	−	0.937531i	$-0.613106\pi$
$601$	−17.0578	−0.695803	−0.347901	+	0.937531i	$0.613106\pi$
$602$	2.21953	0.0904612
$603$	11.5128	0.468835
$604$	−19.2777	−0.784400
$605$	0	0
$606$	−0.527867	−0.0214431
$607$	18.7376	0.760536	0.380268	−	0.924876i	$-0.375832\pi$
$607$	18.7376	0.760536	0.380268	+	0.924876i	$0.375832\pi$
$608$	2.69547	0.109316
$609$	−3.35828	−0.136084
$610$	−1.66598	−0.0674535
$611$	−1.00884	−0.0408133
$612$	−2.23341	−0.0902803
$613$	22.7598	0.919259	0.459629	−	0.888111i	$-0.347982\pi$
$613$	22.7598	0.919259	0.459629	+	0.888111i	$0.347982\pi$
$614$	15.1612	0.611858
$615$	0.976971	0.0393953
$616$	0	0
$617$	−2.57624	−0.103715	−0.0518577	−	0.998654i	$-0.516514\pi$
$617$	−2.57624	−0.103715	−0.0518577	+	0.998654i	$0.516514\pi$
$618$	14.2296	0.572398
$619$	−27.6314	−1.11060	−0.555300	−	0.831650i	$-0.687397\pi$
$619$	−27.6314	−1.11060	−0.555300	+	0.831650i	$0.687397\pi$
$620$	−1.14243	−0.0458812
$621$	−35.0216	−1.40537
$622$	1.98848	0.0797307
$623$	5.70990	0.228762
$624$	−0.116133	−0.00464905
$625$	24.7831	0.991324
$626$	−20.1505	−0.805375
$627$	0	0
$628$	5.36233	0.213980
$629$	9.39539	0.374619
$630$	0.391986	0.0156171
$631$	33.2452	1.32347	0.661735	−	0.749738i	$-0.269820\pi$
$631$	33.2452	1.32347	0.661735	+	0.749738i	$0.269820\pi$
$632$	4.75388	0.189099
$633$	10.2140	0.405972
$634$	−16.2445	−0.645150
$635$	1.05160	0.0417315
$636$	−9.18427	−0.364180
$637$	0.646179	0.0256025
$638$	0	0
$639$	−24.6729	−0.976046
$640$	0.120305	0.00475546
$641$	18.1615	0.717338	0.358669	−	0.933465i	$-0.383231\pi$
$641$	18.1615	0.717338	0.358669	+	0.933465i	$0.383231\pi$
$642$	17.4073	0.687010
$643$	30.2467	1.19281	0.596407	−	0.802682i	$-0.296595\pi$
$643$	30.2467	1.19281	0.596407	+	0.802682i	$0.296595\pi$
$644$	−11.1504	−0.439387
$645$	−0.160252	−0.00630993
$646$	2.69547	0.106052
$647$	−16.6447	−0.654372	−0.327186	−	0.944960i	$-0.606100\pi$
$647$	−16.6447	−0.654372	−0.327186	+	0.944960i	$0.606100\pi$
$648$	−2.68836	−0.105609
$649$	0	0
$650$	−0.661281	−0.0259376
$651$	−12.1297	−0.475399
$652$	−8.15707	−0.319456
$653$	−15.3622	−0.601169	−0.300584	−	0.953755i	$-0.597182\pi$
$653$	−15.3622	−0.601169	−0.300584	+	0.953755i	$0.597182\pi$
$654$	−16.2723	−0.636298
$655$	0.348999	0.0136365
$656$	−9.27509	−0.362131
$657$	3.11966	0.121710
$658$	11.0960	0.432568
$659$	−9.66950	−0.376670	−0.188335	−	0.982105i	$-0.560309\pi$
$659$	−9.66950	−0.376670	−0.188335	+	0.982105i	$0.560309\pi$
$660$	0	0
$661$	−50.0393	−1.94630	−0.973152	−	0.230164i	$-0.926074\pi$
$661$	−50.0393	−1.94630	−0.973152	+	0.230164i	$0.926074\pi$
$662$	16.0144	0.622418
$663$	−0.116133	−0.00451024
$664$	12.7972	0.496628
$665$	−0.473082	−0.0183453
$666$	20.9838	0.813105
$667$	20.0949	0.778079
$668$	−8.23508	−0.318625
$669$	3.72899	0.144171
$670$	−0.620144	−0.0239583
$671$	0	0
$672$	1.27732	0.0492738
$673$	1.65699	0.0638721	0.0319361	−	0.999490i	$-0.489833\pi$
$673$	1.65699	0.0638721	0.0319361	+	0.999490i	$0.489833\pi$
$674$	1.23076	0.0474070
$675$	22.8443	0.879276
$676$	−12.9824	−0.499323
$677$	38.9964	1.49876	0.749378	−	0.662143i	$-0.230353\pi$
$677$	38.9964	1.49876	0.749378	+	0.662143i	$0.230353\pi$
$678$	16.0636	0.616919
$679$	10.9483	0.420158
$680$	0.120305	0.00461347
$681$	17.2815	0.662230
$682$	0	0
$683$	−3.38675	−0.129590	−0.0647952	−	0.997899i	$-0.520639\pi$
$683$	−3.38675	−0.129590	−0.0647952	+	0.997899i	$0.520639\pi$
$684$	6.02009	0.230184
$685$	2.36596	0.0903986
$686$	−17.3193	−0.661256
$687$	0.502141	0.0191579
$688$	1.52139	0.0580025
$689$	1.39136	0.0530065
$690$	0.805071	0.0306485
$691$	−10.6459	−0.404990	−0.202495	−	0.979283i	$-0.564905\pi$
$691$	−10.6459	−0.404990	−0.202495	+	0.979283i	$0.564905\pi$
$692$	−4.06308	−0.154455
$693$	0	0
$694$	21.1460	0.802690
$695$	0.222334	0.00843361
$696$	−2.30196	−0.0872554
$697$	−9.27509	−0.351319
$698$	9.48933	0.359176
$699$	9.24770	0.349780
$700$	7.27329	0.274904
$701$	36.6666	1.38488	0.692438	−	0.721477i	$-0.256536\pi$
$701$	36.6666	1.38488	0.692438	+	0.721477i	$0.256536\pi$
$702$	−0.607772	−0.0229389
$703$	−25.3250	−0.955150
$704$	0	0
$705$	−0.801145	−0.0301729
$706$	−1.34305	−0.0505464
$707$	−0.879555	−0.0330791
$708$	0.568014	0.0213473
$709$	−0.0331559	−0.00124520	−0.000622599	−	1.00000i	$-0.500198\pi$
$709$	−0.0331559	−0.00124520	−0.000622599		1.00000i	$0.500198\pi$
$710$	1.32903	0.0498776
$711$	10.6174	0.398183
$712$	3.91389	0.146679
$713$	72.5803	2.71815
$714$	1.27732	0.0478026
$715$	0	0
$716$	7.12558	0.266295
$717$	−17.1233	−0.639480
$718$	25.7041	0.959270
$719$	13.4386	0.501173	0.250587	−	0.968094i	$-0.419376\pi$
$719$	13.4386	0.501173	0.250587	+	0.968094i	$0.419376\pi$
$720$	0.268690	0.0100135
$721$	23.7100	0.883005
$722$	11.7344	0.436711
$723$	17.5950	0.654366
$724$	7.33962	0.272775
$725$	−13.1077	−0.486808
$726$	0	0
$727$	33.5724	1.24513	0.622565	−	0.782568i	$-0.286091\pi$
$727$	33.5724	1.24513	0.622565	+	0.782568i	$0.286091\pi$
$728$	−0.193506	−0.00717181
$729$	6.03139	0.223385
$730$	−0.168043	−0.00621957
$731$	1.52139	0.0562707
$732$	−12.1246	−0.448139
$733$	11.9964	0.443098	0.221549	−	0.975149i	$-0.428889\pi$
$733$	11.9964	0.443098	0.221549	+	0.975149i	$0.428889\pi$
$734$	−25.0115	−0.923193
$735$	0.513146	0.0189277
$736$	−7.64311	−0.281729
$737$	0	0
$738$	−20.7151	−0.762533
$739$	29.4418	1.08304	0.541518	−	0.840689i	$-0.317850\pi$
$739$	29.4418	1.08304	0.541518	+	0.840689i	$0.317850\pi$
$740$	−1.13031	−0.0415510
$741$	0.313033	0.0114996
$742$	−15.3032	−0.561800
$743$	0.681298	0.0249944	0.0124972	−	0.999922i	$-0.496022\pi$
$743$	0.681298	0.0249944	0.0124972	+	0.999922i	$0.496022\pi$
$744$	−8.31437	−0.304819
$745$	0.343541	0.0125864
$746$	5.43001	0.198807
$747$	28.5814	1.04574
$748$	0	0
$749$	29.0048	1.05981
$750$	−1.05180	−0.0384064
$751$	14.7614	0.538650	0.269325	−	0.963049i	$-0.413199\pi$
$751$	14.7614	0.538650	0.269325	+	0.963049i	$0.413199\pi$
$752$	7.60585	0.277357
$753$	−2.49572	−0.0909492
$754$	0.348731	0.0127000
$755$	2.31920	0.0844044
$756$	6.68476	0.243122
$757$	16.0094	0.581870	0.290935	−	0.956743i	$-0.406034\pi$
$757$	16.0094	0.581870	0.290935	+	0.956743i	$0.406034\pi$
$758$	3.69485	0.134203
$759$	0	0
$760$	−0.324278	−0.0117628
$761$	−29.8594	−1.08240	−0.541200	−	0.840894i	$-0.682030\pi$
$761$	−29.8594	−1.08240	−0.541200	+	0.840894i	$0.682030\pi$
$762$	7.65331	0.277250
$763$	−27.1137	−0.981580
$764$	−8.04902	−0.291203
$765$	0.268690	0.00971451
$766$	2.40475	0.0868872
$767$	−0.0860504	−0.00310710
$768$	0.875550	0.0315937
$769$	20.6198	0.743570	0.371785	−	0.928319i	$-0.378746\pi$
$769$	20.6198	0.743570	0.371785	+	0.928319i	$0.378746\pi$
$770$	0	0
$771$	−17.7638	−0.639749
$772$	11.8523	0.426574
$773$	−43.4066	−1.56123	−0.780614	−	0.625013i	$-0.785094\pi$
$773$	−43.4066	−1.56123	−0.780614	+	0.625013i	$0.785094\pi$
$774$	3.39789	0.122135
$775$	−47.3434	−1.70062
$776$	7.50460	0.269400
$777$	−12.0009	−0.430532
$778$	27.8761	0.999407
$779$	25.0007	0.895743
$780$	0.0139714	0.000500255 0
$781$	0	0
$782$	−7.64311	−0.273317
$783$	−12.0471	−0.430528
$784$	−4.87167	−0.173988
$785$	−0.645114	−0.0230251
$786$	2.53994	0.0905966
$787$	−33.1617	−1.18209	−0.591043	−	0.806640i	$-0.701284\pi$
$787$	−33.1617	−1.18209	−0.591043	+	0.806640i	$0.701284\pi$
$788$	8.46442	0.301533
$789$	18.2689	0.650390
$790$	−0.571914	−0.0203478
$791$	26.7659	0.951685
$792$	0	0
$793$	1.83680	0.0652267
$794$	20.1130	0.713785
$795$	1.10491	0.0391872
$796$	−3.17230	−0.112439
$797$	14.0388	0.497278	0.248639	−	0.968596i	$-0.420017\pi$
$797$	14.0388	0.497278	0.248639	+	0.968596i	$0.420017\pi$
$798$	−3.44298	−0.121880
$799$	7.60585	0.269076
$800$	4.98553	0.176265
$801$	8.74133	0.308860
$802$	6.79434	0.239916
$803$	0	0
$804$	−4.51327	−0.159171
$805$	1.34144	0.0472797
$806$	1.25957	0.0443665
$807$	14.0115	0.493227
$808$	−0.602898	−0.0212099
$809$	−9.43393	−0.331679	−0.165840	−	0.986153i	$-0.553033\pi$
$809$	−9.43393	−0.331679	−0.165840	+	0.986153i	$0.553033\pi$
$810$	0.323423	0.0113639
$811$	20.8614	0.732542	0.366271	−	0.930508i	$-0.380634\pi$
$811$	20.8614	0.732542	0.366271	+	0.930508i	$0.380634\pi$
$812$	−3.83562	−0.134604
$813$	19.8927	0.697669
$814$	0	0
$815$	0.981334	0.0343746
$816$	0.875550	0.0306504
$817$	−4.10086	−0.143471
$818$	−16.8723	−0.589927
$819$	−0.432179	−0.0151015
$820$	1.11584	0.0389667
$821$	−31.7661	−1.10864	−0.554322	−	0.832302i	$-0.687022\pi$
$821$	−31.7661	−1.10864	−0.554322	+	0.832302i	$0.687022\pi$
$822$	17.2189	0.600578
$823$	−5.50343	−0.191838	−0.0959188	−	0.995389i	$-0.530579\pi$
$823$	−5.50343	−0.191838	−0.0959188	+	0.995389i	$0.530579\pi$
$824$	16.2522	0.566171
$825$	0	0
$826$	0.946450	0.0329312
$827$	−6.51122	−0.226417	−0.113209	−	0.993571i	$-0.536113\pi$
$827$	−6.51122	−0.226417	−0.113209	+	0.993571i	$0.536113\pi$
$828$	−17.0702	−0.593231
$829$	6.56777	0.228108	0.114054	−	0.993475i	$-0.463616\pi$
$829$	6.56777	0.228108	0.114054	+	0.993475i	$0.463616\pi$
$830$	−1.53956	−0.0534390
$831$	−10.9911	−0.381276
$832$	−0.132640	−0.00459847
$833$	−4.87167	−0.168793
$834$	1.61810	0.0560301
$835$	0.990719	0.0342852
$836$	0	0
$837$	−43.5125	−1.50401
$838$	19.9568	0.689396
$839$	−32.8287	−1.13337	−0.566687	−	0.823933i	$-0.691775\pi$
$839$	−32.8287	−1.13337	−0.566687	+	0.823933i	$0.691775\pi$
$840$	−0.153668	−0.00530205
$841$	−22.0875	−0.761640
$842$	4.17192	0.143774
$843$	7.00374	0.241222
$844$	11.6659	0.401555
$845$	1.56184	0.0537291
$846$	16.9870	0.584025
$847$	0	0
$848$	−10.4897	−0.360218
$849$	23.2826	0.799056
$850$	4.98553	0.171002
$851$	71.8101	2.46162
$852$	9.67237	0.331370
$853$	17.2174	0.589513	0.294756	−	0.955572i	$-0.404762\pi$
$853$	17.2174	0.589513	0.294756	+	0.955572i	$0.404762\pi$
$854$	−20.2026	−0.691318
$855$	−0.724245	−0.0247687
$856$	19.8815	0.679537
$857$	−3.96083	−0.135300	−0.0676498	−	0.997709i	$-0.521550\pi$
$857$	−3.96083	−0.135300	−0.0676498	+	0.997709i	$0.521550\pi$
$858$	0	0
$859$	−11.6086	−0.396079	−0.198040	−	0.980194i	$-0.563457\pi$
$859$	−11.6086	−0.396079	−0.198040	+	0.980194i	$0.563457\pi$
$860$	−0.183031	−0.00624129
$861$	11.8473	0.403754
$862$	3.11158	0.105981
$863$	19.7991	0.673970	0.336985	−	0.941510i	$-0.390593\pi$
$863$	19.7991	0.673970	0.336985	+	0.941510i	$0.390593\pi$
$864$	4.58211	0.155887
$865$	0.488808	0.0166200
$866$	−27.7967	−0.944571
$867$	0.875550	0.0297352
$868$	−13.8538	−0.470227
$869$	0	0
$870$	0.276936	0.00938902
$871$	0.683731	0.0231673
$872$	−18.5853	−0.629376
$873$	16.7609	0.567269
$874$	20.6018	0.696866
$875$	−1.75256	−0.0592474
$876$	−1.22298	−0.0413207
$877$	−32.3111	−1.09107	−0.545533	−	0.838089i	$-0.683673\pi$
$877$	−32.3111	−1.09107	−0.545533	+	0.838089i	$0.683673\pi$
$878$	17.0281	0.574670
$879$	−26.3924	−0.890193
$880$	0	0
$881$	58.1195	1.95810	0.979048	−	0.203631i	$-0.0652743\pi$
$881$	58.1195	1.95810	0.979048	+	0.203631i	$0.0652743\pi$
$882$	−10.8804	−0.366364
$883$	−20.9990	−0.706672	−0.353336	−	0.935496i	$-0.614953\pi$
$883$	−20.9990	−0.706672	−0.353336	+	0.935496i	$0.614953\pi$
$884$	−0.132640	−0.00446117
$885$	−0.0683348	−0.00229705
$886$	−22.8935	−0.769123
$887$	5.45289	0.183090	0.0915451	−	0.995801i	$-0.470819\pi$
$887$	5.45289	0.183090	0.0915451	+	0.995801i	$0.470819\pi$
$888$	−8.22614	−0.276051
$889$	12.7523	0.427698
$890$	−0.470859	−0.0157832
$891$	0	0
$892$	4.25903	0.142603
$893$	−20.5013	−0.686051
$894$	2.50021	0.0836197
$895$	−0.857241	−0.0286544
$896$	1.45888	0.0487378
$897$	−0.887619	−0.0296367
$898$	10.1036	0.337163
$899$	24.9669	0.832692
$900$	11.1347	0.371158
$901$	−10.4897	−0.349463
$902$	0	0
$903$	−1.94331	−0.0646692
$904$	18.3469	0.610208
$905$	−0.882991	−0.0293516
$906$	16.8786	0.560755
$907$	44.9806	1.49356	0.746778	−	0.665074i	$-0.231600\pi$
$907$	44.9806	1.49356	0.746778	+	0.665074i	$0.231600\pi$
$908$	19.7379	0.655026
$909$	−1.34652	−0.0446612
$910$	0.0232797	0.000771714 0
$911$	7.38148	0.244559	0.122280	−	0.992496i	$-0.460980\pi$
$911$	7.38148	0.244559	0.122280	+	0.992496i	$0.460980\pi$
$912$	−2.36002	−0.0781480
$913$	0	0
$914$	−41.7070	−1.37954
$915$	1.45865	0.0482214
$916$	0.573514	0.0189494
$917$	4.23215	0.139758
$918$	4.58211	0.151232
$919$	−21.6230	−0.713277	−0.356639	−	0.934242i	$-0.616077\pi$
$919$	−21.6230	−0.713277	−0.356639	+	0.934242i	$0.616077\pi$
$920$	0.919503	0.0303151
$921$	−13.2744	−0.437407
$922$	−12.0145	−0.395675
$923$	−1.46530	−0.0482310
$924$	0	0
$925$	−46.8410	−1.54012
$926$	19.0604	0.626364
$927$	36.2978	1.19218
$928$	−2.62915	−0.0863062
$929$	−1.00816	−0.0330768	−0.0165384	−	0.999863i	$-0.505265\pi$
$929$	−1.00816	−0.0330768	−0.0165384	+	0.999863i	$0.505265\pi$
$930$	1.00026	0.0327997
$931$	13.1314	0.430365
$932$	10.5622	0.345975
$933$	−1.74101	−0.0569982
$934$	4.92280	0.161079
$935$	0	0
$936$	−0.296240	−0.00968291
$937$	60.5898	1.97938	0.989691	−	0.143216i	$-0.0457444\pi$
$937$	60.5898	1.97938	0.989691	+	0.143216i	$0.0457444\pi$
$938$	−7.52021	−0.245544
$939$	17.6428	0.575750
$940$	−0.915019	−0.0298446
$941$	21.4283	0.698544	0.349272	−	0.937021i	$-0.386429\pi$
$941$	21.4283	0.698544	0.349272	+	0.937021i	$0.386429\pi$
$942$	−4.69499	−0.152971
$943$	−70.8905	−2.30851
$944$	0.648751	0.0211150
$945$	−0.804208	−0.0261609
$946$	0	0
$947$	23.8595	0.775330	0.387665	−	0.921800i	$-0.373282\pi$
$947$	23.8595	0.775330	0.387665	+	0.921800i	$0.373282\pi$
$948$	−4.16226	−0.135184
$949$	0.185274	0.00601424
$950$	−13.4383	−0.435997
$951$	14.2228	0.461207
$952$	1.45888	0.0472826
$953$	−14.1985	−0.459935	−0.229968	−	0.973198i	$-0.573862\pi$
$953$	−14.1985	−0.459935	−0.229968	+	0.973198i	$0.573862\pi$
$954$	−23.4279	−0.758505
$955$	0.968335	0.0313346
$956$	−19.5572	−0.632524
$957$	0	0
$958$	31.8201	1.02806
$959$	28.6909	0.926477
$960$	−0.105333	−0.00339960
$961$	59.1771	1.90894
$962$	1.24621	0.0401793
$963$	44.4036	1.43089
$964$	20.0960	0.647248
$965$	−1.42589	−0.0459009
$966$	9.76273	0.314111
$967$	−17.7926	−0.572170	−0.286085	−	0.958204i	$-0.592354\pi$
$967$	−17.7926	−0.572170	−0.286085	+	0.958204i	$0.592354\pi$
$968$	0	0
$969$	−2.36002	−0.0758147
$970$	−0.902839	−0.0289884
$971$	58.6943	1.88359	0.941795	−	0.336189i	$-0.109138\pi$
$971$	58.6943	1.88359	0.941795	+	0.336189i	$0.109138\pi$
$972$	16.1001	0.516412
$973$	2.69614	0.0864344
$974$	−16.4732	−0.527834
$975$	0.578985	0.0185424
$976$	−13.8480	−0.443263
$977$	29.4852	0.943314	0.471657	−	0.881782i	$-0.343656\pi$
$977$	29.4852	0.943314	0.471657	+	0.881782i	$0.343656\pi$
$978$	7.14193	0.228374
$979$	0	0
$980$	0.586085	0.0187218
$981$	−41.5085	−1.32527
$982$	35.2629	1.12528
$983$	−59.1836	−1.88766	−0.943832	−	0.330427i	$-0.892807\pi$
$983$	−59.1836	−1.88766	−0.943832	+	0.330427i	$0.892807\pi$
$984$	8.12080	0.258882
$985$	−1.01831	−0.0324460
$986$	−2.62915	−0.0837293
$987$	−9.71512	−0.309236
$988$	0.357527	0.0113745
$989$	11.6282	0.369754
$990$	0	0
$991$	−17.2675	−0.548519	−0.274260	−	0.961656i	$-0.588433\pi$
$991$	−17.2675	−0.548519	−0.274260	+	0.961656i	$0.588433\pi$
$992$	−9.49616	−0.301503
$993$	−14.0214	−0.444957
$994$	16.1165	0.511185
$995$	0.381643	0.0120989
$996$	−11.2046	−0.355031
$997$	−44.2057	−1.40001	−0.700004	−	0.714139i	$-0.746818\pi$
$997$	−44.2057	−1.40001	−0.700004	+	0.714139i	$0.746818\pi$
$998$	31.9481	1.01130
$999$	−43.0508	−1.36207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4114.2.a.bg.1.6		8
11.2	odd	10		374.2.g.f.103.3	yes	16
11.6	odd	10		374.2.g.f.69.3	✓	16
11.10	odd	2		4114.2.a.bi.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
374.2.g.f.69.3	✓	16	11.6	odd	10
374.2.g.f.103.3	yes	16	11.2	odd	10
4114.2.a.bg.1.6		8	1.1	even	1	trivial
4114.2.a.bi.1.6		8	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4114 = 2 \cdot 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4114.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.875550 q^{3} +1.00000 q^{4} -0.120305 q^{5} -0.875550 q^{6} -1.45888 q^{7} -1.00000 q^{8} -2.23341 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.875550 q^{3} +1.00000 q^{4} -0.120305 q^{5} -0.875550 q^{6} -1.45888 q^{7} -1.00000 q^{8} -2.23341 q^{9} +0.120305 q^{10} +0.875550 q^{12} -0.132640 q^{13} +1.45888 q^{14} -0.105333 q^{15} +1.00000 q^{16} +1.00000 q^{17} +2.23341 q^{18} -2.69547 q^{19} -0.120305 q^{20} -1.27732 q^{21} +7.64311 q^{23} -0.875550 q^{24} -4.98553 q^{25} +0.132640 q^{26} -4.58211 q^{27} -1.45888 q^{28} +2.62915 q^{29} +0.105333 q^{30} +9.49616 q^{31} -1.00000 q^{32} -1.00000 q^{34} +0.175510 q^{35} -2.23341 q^{36} +9.39539 q^{37} +2.69547 q^{38} -0.116133 q^{39} +0.120305 q^{40} -9.27509 q^{41} +1.27732 q^{42} +1.52139 q^{43} +0.268690 q^{45} -7.64311 q^{46} +7.60585 q^{47} +0.875550 q^{48} -4.87167 q^{49} +4.98553 q^{50} +0.875550 q^{51} -0.132640 q^{52} -10.4897 q^{53} +4.58211 q^{54} +1.45888 q^{56} -2.36002 q^{57} -2.62915 q^{58} +0.648751 q^{59} -0.105333 q^{60} -13.8480 q^{61} -9.49616 q^{62} +3.25828 q^{63} +1.00000 q^{64} +0.0159572 q^{65} -5.15478 q^{67} +1.00000 q^{68} +6.69193 q^{69} -0.175510 q^{70} +11.0472 q^{71} +2.23341 q^{72} -1.39682 q^{73} -9.39539 q^{74} -4.36508 q^{75} -2.69547 q^{76} +0.116133 q^{78} -4.75388 q^{79} -0.120305 q^{80} +2.68836 q^{81} +9.27509 q^{82} -12.7972 q^{83} -1.27732 q^{84} -0.120305 q^{85} -1.52139 q^{86} +2.30196 q^{87} -3.91389 q^{89} -0.268690 q^{90} +0.193506 q^{91} +7.64311 q^{92} +8.31437 q^{93} -7.60585 q^{94} +0.324278 q^{95} -0.875550 q^{96} -7.50460 q^{97} +4.87167 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9	\( 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9} - 2 q^{10} - 5 q^{12} + 5 q^{13} + 11 q^{14} + 2 q^{15} + 8 q^{16} + 8 q^{17} - 11 q^{18} - 15 q^{19} + 2 q^{20} - 12 q^{23} + 5 q^{24} + 24 q^{25} - 5 q^{26} - 17 q^{27} - 11 q^{28} - 22 q^{29} - 2 q^{30} - 7 q^{31} - 8 q^{32} - 8 q^{34} - 10 q^{35} + 11 q^{36} + 15 q^{37} + 15 q^{38} - 35 q^{39} - 2 q^{40} - 17 q^{41} - 8 q^{43} + 3 q^{45} + 12 q^{46} + 18 q^{47} - 5 q^{48} + q^{49} - 24 q^{50} - 5 q^{51} + 5 q^{52} + 20 q^{53} + 17 q^{54} + 11 q^{56} - 9 q^{57} + 22 q^{58} + 6 q^{59} + 2 q^{60} + 5 q^{61} + 7 q^{62} + 14 q^{63} + 8 q^{64} - 3 q^{65} + 13 q^{67} + 8 q^{68} + 46 q^{69} + 10 q^{70} - 18 q^{71} - 11 q^{72} + 4 q^{73} - 15 q^{74} - 49 q^{75} - 15 q^{76} + 35 q^{78} - 21 q^{79} + 2 q^{80} + 8 q^{81} + 17 q^{82} - 38 q^{83} + 2 q^{85} + 8 q^{86} + 46 q^{87} + 12 q^{89} - 3 q^{90} - 5 q^{91} - 12 q^{92} + 3 q^{93} - 18 q^{94} - 63 q^{95} + 5 q^{96} - 66 q^{97} - q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9 - 2 * q^10 - 5 * q^12 + 5 * q^13 + 11 * q^14 + 2 * q^15 + 8 * q^16 + 8 * q^17 - 11 * q^18 - 15 * q^19 + 2 * q^20 - 12 * q^23 + 5 * q^24 + 24 * q^25 - 5 * q^26 - 17 * q^27 - 11 * q^28 - 22 * q^29 - 2 * q^30 - 7 * q^31 - 8 * q^32 - 8 * q^34 - 10 * q^35 + 11 * q^36 + 15 * q^37 + 15 * q^38 - 35 * q^39 - 2 * q^40 - 17 * q^41 - 8 * q^43 + 3 * q^45 + 12 * q^46 + 18 * q^47 - 5 * q^48 + q^49 - 24 * q^50 - 5 * q^51 + 5 * q^52 + 20 * q^53 + 17 * q^54 + 11 * q^56 - 9 * q^57 + 22 * q^58 + 6 * q^59 + 2 * q^60 + 5 * q^61 + 7 * q^62 + 14 * q^63 + 8 * q^64 - 3 * q^65 + 13 * q^67 + 8 * q^68 + 46 * q^69 + 10 * q^70 - 18 * q^71 - 11 * q^72 + 4 * q^73 - 15 * q^74 - 49 * q^75 - 15 * q^76 + 35 * q^78 - 21 * q^79 + 2 * q^80 + 8 * q^81 + 17 * q^82 - 38 * q^83 + 2 * q^85 + 8 * q^86 + 46 * q^87 + 12 * q^89 - 3 * q^90 - 5 * q^91 - 12 * q^92 + 3 * q^93 - 18 * q^94 - 63 * q^95 + 5 * q^96 - 66 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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